{
  "title": "Canonical transport from countable cone tests",
  "license": "CC0-1.0",
  "proof": "../../OA-FLOW-MSF.html#oa-flow.msf.constructive",
  "parameter": "A fixed x; all data and index choices are measurable in x.",
  "types": {
    "W_x": "H_x -> K_x",
    "p_j": "P_x subset H_x",
    "q_k": "Q_x subset K_x"
  },
  "residual": "D_jk = sup_l |<theta(b_l) q_k,q_k> - <b_l p_j,p_j>|",
  "residual_identity": "D_jk = ||omega_q_k - omega_W_x_p_j||",
  "cone_error": "D_jk < epsilon^2 implies ||q_k-W_x p_j|| < epsilon",
  "frame_choice": "a_t = sum_j r_tj p_j; ||a_t-e_i|| < 2^(-m)",
  "coefficient_sum": "R_t = sum_j |r_tj|",
  "cone_tolerance": "epsilon = 2^(-m)/(1+R_t)",
  "frame_image": "h_i,m = sum_j r_tj q_k_j",
  "frame_error": "||h_i,m-W_x e_i|| < R_t epsilon + 2^(-m) < 2^(1-m)",
  "scope": "Countable exact norm tests; no finite decision procedure is asserted.",
  "sample": {
    "m": 3,
    "R_t": 3,
    "epsilon": "1/32",
    "cone_contribution": "3/32",
    "source_contribution": "1/8",
    "sum": "7/32",
    "strict_bound": "1/4"
  }
}